To be updated until april 30th

Médéric Argentina

## Scaling the viscous Hydraulic Jump

Médéric Argentina
mederic.argentina@univ-cotedazur.fr
Institut de Physique de Nice, Nice
The formation mechanism of hydraulic jumps has been proposed by Belanger in 1828 and rationalised by Lord Rayleigh in 1914. As the Froude number becomes higher than one, the flow super criticality induces an instability which yields the emergence of a steep structure at the fluid surface. Strongly deformed liquid-air interface can be observed as a jet of viscous fluid impinges a flat boundary at high enough velocity. In this experimental setup, the location of the jump depends on the viscosity of the liquid, as shown by T. Bohr et al. in 1997. In 2014, A. Duchesne et al. have established the constancy of the Froude number at jump. Hence, it remains a contradiction, in which the radial hydraulic jump location might be explained through inviscid theory, but is also viscosity dependent. We present a model based on the 2011 Rojas et al. PRL, which solves this paradox. The agreement with experimental measurements is excellent not only for the prediction of the position of the hydraulic jump, but also for the determination of the fluid thickness profile. We predict theoretically the critical value of the Froude number, which matches perfectly to that measured by Duchesne et al.

Nicolas Cellier

## scikit-fdiff, a new tool for PDE solving

Nicolas Cellier
contact@nicolas-cellier.net
Université Savoie Mont-Blanc

Scikit-FDiff (formerly known as Triflow) is a new tool, written in pure Python, that focus on reducing the time between the developpement of the mathematical model and the numerical solving. It allows an easy and automatic finite difference discretization, thanks to a symbolic processing that can deal with systems of multi-dimensional partial differential equation with complex boundary conditions.

Using finite differences and the method of lines, it allows the transformation of the original PDE into an ODE, providing a fast computation of the temporal evolution vector and the Jacobian matrix. The later is pre-computed in a symbolic way and sparse by nature. It can be evaluated with as few computational resources as possible, allowing the use of implicit and explicit solvers at a reasonable cost.

Classic ODE solvers have been implemented (or made available from dedicated python libraries), including backward and forward Euler scheme, Crank-Nickolson, explicit Runge-Kutta. More complexes ones, like improved Rosenbrock-Wanner schemes up to the 6th order, are also available. The time-step is managed by a built-in error computation, which ensures the accuracy of the solution. The main goal of the software is to minimize the time spent writting numerical solvers to focus on model development and data analysis.

Scikit-Fdiff is then able to solve toy cases in a few line of code as well as complex models. Extra tools are available, such as data saving during the simulation, real-time plotting and post-processing. It has been validated with the shallow-water equation on dam-breaks and the steady-lake case. It has also been applied to heated falling-films, dropplet spread and simple moisture flow in porous medium.

Amin Chabchoub

## Nonlinear, short-crested and localized waves

Amin Chabchoub
amin.chabchoub@sydney.edu.au
Centre for Wind, Waves and Water, School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia
Solitons and breathers are known to model stationary and extreme localizations in nonlinear dispersive media. Indeed, a series of laboratory experiments, for instance in water waves, optics and BEC, confirmed the validity of the the uni-directional nonlinear Schrödinger equation (NLSE) to describe the spatio-temporal dynamics of such waves. In this study, we report observations of slanted, and thus, directional localized envelope soliton and breather dynamics in a water wave basin. The water surface displacement has been stereo-reconstructed using a marker-net, deployed at the center of the basin, and two synchronized high-speed cameras. The results are in very good agreement with the hyperbolic 2D+1 NLS predictions and confirm for the first time that short-crested as well as slanted strongly-localized waves can be also described by a simplified nonlinear wave framework.

Didier Clamond

## On an improved model for long internal gravity waves

Didier Clamond
didierc@unice.fr
Université Côte d'Azur, LJAD, Parc Valrose, 06108 Nice cedex 2
We consider 2D irrotational flows of incompressible fluids stratified in two homogeneous shallow layers, bounded below by a horizontal impermeable bottom and above by a rigid lid. Weakly dispersive fully nonlinear equations of Serre-Green-Nagdhi type are often used to model long internal waves. We show how to improve these equations with better dispersive properties and reduced Kelvin-Helmholtz instability, while remaining asymptotically consistent and keeping all the conservation laws.

Giovanni Dematteis

## Experimental evidence of hydrodynamic instantons and their unifying role in the theory of rogue waves

Giovanni Dematteis
giovannidematteis@gmail.com
Università degli Studi di Torino
We interpret the formation of rogue waves in a wave flume in light of tools of large deviation theory. By numerical optimization we compute the instantons of the problem, i.e. the most likely realizations leading to extreme surface elevations via the governing nonlinear Schroedinger dynamics. We show that strikingly the typical extreme events of the experiment closely follow the instanton evolution, with small fluctuations around it. This is true accross all of the explored forcing regimes, unifying and extending the existing limiting results for the linear and highly nonlinear cases.

Benjamin Dollet

## Interfacial nonlinearities to damp sloshing waves

Benjamin Dollet
benjamin.dollet@univ-grenoble-alpes.fr
Laboratoire Intredisciplinaire de Physique, UMR 5588 CNRs/Université Grenoble Alpes, 140 rue de la Physique, 38402 Saint-Martin-d'Hères, France
Sloshing describes the oscillations of liquids in reservoirs. It is often detrimental and can lead e.g. to coffee spilling, or to destabilisation of tankers and spacecrafts, especially in its large-amplitude, nonlinear regimes. Therefore, understanding and optimising its damping is of primary importance for applications. Presenting experimental measurements and the associated theoreticaI modeling, I will discuss two ways to increase sloshing damping by interfacial effects: either using a foam layer, or, in the case of partial wetting, tuning the contact angle hysteresis. Interestingly, these two strategies lead to novel nonlinearities which, contrary to the usual large-amplitude effects, manifest themselves all the most that sloshing amplitude is small, leading to singularities like the finite-time arrest of the oscillations of the liquid/air interface.

Alexis Duchesne

## Birth of a hydraulic jump

Alexis Duchesne
alexis.duchesne@univ-lille.fr
IIEMN, AIMAN-FILMS, UMR CNRS 8520, Cité Scientifique, Avenue Poincaré, 59652 VILLENEUVE D’ASCQ CEDEX, France

Alexis Duchesne, Tomas Bohr and Anders Andersen

The hydraulic jump, i.e., the sharp transition between a supercritical and a subcritical free-surface flow, has been extensively studied. However, an important question has been left unanswered: How does a hydraulic jump form? We present here an experimental and theoretical study of the formation of stationary hydraulic jumps in centimeter-sized channels.

We start with an empty channel and then change the flow rate abruptly from zero to a constant value. This leads to the formation of a stationary hydraulic jump in a two stage process. Firstly, the channel fills quickly ($\sim 1$ s). Initially the liquid layer shows a linearly increasing height profile and a front position with a square root dependence on time. When the height of the liquid front reaches a critical value, it remains constant throughout the rest of the filling process. At low flow rate the jump forms during the filling of the channel whereas the jump appears at a later stage when the flow rate is high. Secondly, the influence of the downstream boundary condition makes the jump move slowly ($\sim 10$ s) upstream to its final position with exponentially decreasing speed.

Eric Falcon

## Hypergravity Wave Turbulence

Eric Falcon
eric.falcon@univ-paris-diderot.fr
Université Paris Diderot, MSC, CNRS, F-75013 Paris, France
Wave turbulence occurs in various domains of physics (plasma physics, elasticity, or fluid mechanics) but is far to be completely understood, notably for ocean surface waves. By using a large-diameter centrifuge, we are able to tune the gravity field up to 20 times the Earth acceleration. This new technique then allows us to report the first observation of gravity wave turbulence on the surface of a fluid in hyper-gravity environment. This is also a unique solution to significantly expands the inertial range of gravity wave turbulence in laboratory. Wave turbulence properties are then reported as function of the gravity level, and we show that the usual energy transfer by nonlinear wave interactions are modified by large-scale container modes.

Benjamin Favier

## Internal gravity waves generated by turbulent flows

Benjamin Favier
favier@irphe.univ-mrs.fr
Aix Marseille University, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille, France

Many geophysical and astrophysical fluids, including planetary atmospheres, stars and oceans, consist of turbulent flows adjacent to stably-stratified fluid layers. Because waves can drive large-scale flows, increase scalar mixing and are sometimes easier to observe than turbulent motions, two important questions for these fluids are: how much energy goes from the turbulence into internal waves in the stable layer? What kind of waves (i.e. what wavenumbers and frequencies) are generated most efficiently?

In this talk we will answer these two questions by presenting a theoretical prediction for the energy flux spectrum of waves generated by turbulent convection and comparing it with results from 3D direct numerical simulations (DNS) of self-organised convective--stably-stratified fluids. We will show that DNS and theory agree well for the range of strong turbulence-strong stratification parameters tested, giving some confidence in the analytical expression for the energy flux spectrum of the waves, which is based on a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region. We hope that our results will help quantify wave generation in geophysical and astrophysical fluids.

Thomas Frisch

## Leidenfrost Effect: The life of a levitating water droplet on a hot vapour layer

Thomas Frisch
thomas.frisch@inphyni.cnrs.fr
Institut de Physique de Nice, Nice

The Leidenfrost effect is a physical phenomenon in which a liquid droplet floats on its own evaporating vapour due to the presence of a hot substrate underneath. This effect was discovered by Johan Leidenfrost in 1771 and investigated by John Tyndall as narrated in his book “Heat: a mode of motion (1875). Leidenfrost droplets constitutes an interesting out of equilibrium system which can be a nice playground for laboratory experiments on capillarity and fluid motion. In my talk, I will review the recent experimental and theoretical studies that we have undergone in our laboratory. I will discuss the behaviour of Leidenfrost droplets in the super-levitation regime [1,2] which takes place for a very small droplet radius and reveals the signature of the end of the lubrication regime. I will also discuss a new technique for generating Leidenfrost droplet at ambient temperature (20 Celcius) by using a low-pressure atmosphere [3]. These droplets could have applications as micro-reactors. Finally, I will expose theoretical and experimental results on Leidenfrost droplets which are confined in a two-dimensional geometry by means of a Hele-Shaw cell [4,5], in particular their oscillations and the dynamics of a growing hole. Finally, I will conclude with some questions on their interface fluctuations when the system is close to the Leidenfrost transition.

[1] Take off of small Leidenfrost droplets, F Celestini, T Frisch, Y Pomeau, Physical review letters 109 (2012)

[2] The Leidenfrost effect: From quasi-spherical droplets to puddles, Y Pomeau, M Le Berre, F Celestini, T Frisch, Comptes Rendus Mecanique 340 (2012)

[3] Room temperature water Leidenfrost droplets, F Celestini, T Frisch, Y Pomeau Soft Matter 9 (2013)

[4]Two-dimensional Leidenfrost droplets in a Hele-Shaw cell, F Celestini, T Frisch, A Cohen, C Raufaste, L Duchemin, Y Pomeau, Physics of Fluids 26 (2014)

[5] Hole growth dynamics in a two-dimensional Leidenfrost droplet, C Raufaste, F Celestini, A Barzyk, T Frisch, Physics of Fluids 27 (2015)

Christophe Josserand

## Singularity turbulence

Christophe Josserand
LadHyX, CNRS & Ecole Polytechnique, UMR 7646, IP Paris, 91128 Palaiseau, France
We will discuss the singularity induced turbulence obtained using a simple model based on the focusing non-linear Schrödinger equation. We observe a transition from a smooth dynamics towards a strong turbulence regime as the control parameters increase. This strong turbulence regime consists of the midst of the singularities of the NLS equations healed by the viscosity. Kolmogorov-like spectra are observed and will be discussed in the context of the cascade phenomenology. Co-authors: Yves Pomeau and Sergio Rica

Sylvain Joubaud

## Energy cascade in internal wave attractors

Sylvain Joubaud
sylvain.joubaud@ens-lyon.fr
Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
Internal gravity waves play an important role in various geophysical flows. Oceans or atmospheres are indeed stratified in density and support the propagation of such waves. They significantly contribute in the mixing of the ocean, the redistribution of energy and momentum in the middle atmosphere or the transport of sediments and plankton. The subsequent mechanism for the energy transfer from large scales of the injected energy to small scales where dissipation occurs is therefore a critical issue in the dynamics of the ocean or the atmosphere, and also an important fundamental question. In this talk, we will focus on the fate of internal gravity waves in a trapezoidal geometry of the confined fluid domain. The peculiar dispersion relation of these waves lead to strong variation of the wave beam upon reflection on a slope. In such a configuration, the focusing of internal waves prevails, leading to convergence of internal wave rays toward closed loops, the internal wave attractors. The high concentration of energy in attractors make them prone to instabilities. We will show that this experimental set-up models a cascade of triadic interactions and provide an efficient energy pathway from global scale motions to small scale overturning events, which induces significant mixing.

Patrice Le Gal

## Resonances of Internal Gravity Waves in Stratified Shear Flows

Patrice Le_Gal
legal@irphe.univ-mrs.fr
IRPHE - Aix Marseille Université - CNRS - Centrale Marseille

P. Le Gal, G. Facchini, J. Chen, S. Le Dizès, M. Le Bars, B. Favier, IRPHE, Aix Marseille Univ., CNRS, Centrale Marseille, France

U. Harlander, I.D. Borcia, Dept. of Aerodynamics and Fluid Mechanics Brandenburg Univ. of Technology, Cottbus, Germany

W. Meng, Dept. of Mechanical Engineering, Univ. of California, Berkeley, CA 94709, USA

We will present here a new instability mechanism that affects the Plane Couette flow and the Plane Poiseuille flow when these flows are stably stratified in density along the vertical direction, i.e. orthogonal to the horizontal shear. Stratified shear flows are ubiquitous in nature and in a geophysical context, we may think to water flows in submarine canyons, to winds in deep valleys, to currents along sea shores or to laminar flows in canals where density stratification can be due to temperature or salinity gradients. Our study is based on two sets of laboratory experiments with salt stratified water flows, on linear stability analyses and on direct numerical simulations. It follows recent investigations of instabilities in stratified rotating or non rotating shear flows: the stratorotational instability [2],[3], the stratified boundary layer instability [4] where it was shown that these instabilities belong to a class of instabilities caused by the resonant interaction of Doppler shifted internal gravity waves. Our laboratory experiments for Plane Couette and Plane Poiseuille flows, based on visualizations and PIV measurements, show in both cases the appearance of braided wave patterns when the experimental parameters, depending on the Reynolds and Froude numbers, are above a threshold. The non linear saturation of the instability leads to a meandering in the horizontal plane arranged in layers stacked along the vertical direction [5]. Comparison with theoretical predictions for the instability threshold and the critical wavenumbers calculated by linear analysis is excellent. Moreover, direct numerical simulations permit to complete the description of this instability that can be interpreted as a resonant interaction of boundary trapped waves [6].

[1] S. Orszag, JFM 50(4), 689-703, 1971.

[2] M. Le Bars & P. Le Gal, Phys. Rev. Lett. 99, 064502, 2007.

[3] G. Rüdiger, T. Seelig, M. Schultz, M. Gellert, Ch. Egbers & U. Harlander, GAFD,111, 429-447, 2017.

[4] J. Chen, Y. Bai, & S. Le Dizès, JFM 795, 262-277, 2016.

[5] D. Lucas, C.P. Caulfield, R. R. Kerswell, arXiv:1808.01178, 2019.

[6] G. Facchini, B. Favier, P. Le Gal, M. Wang, M. Le Bars, JFM 853, 205-234, 2018.

Thomas Le Reun

## Transition from inertial wave turbulence to geostrophic turbulence in rotating fluids - an experimental study

Thomas Le_Reun
lereun@irphe.univ-mrs.fr
Aix-Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille

We present an experimental investigation of the turbulent saturation of the flow driven by parametric resonance of inertial waves in a rotating fluid. In our setup, a half-meter wide ellipsoid filled with water is brought to solid body rotation, and then undergoes sustained harmonic modulation of its rotation rate. This triggers the exponential growth of a pair of inertial waves via a mechanism called the libration-driven elliptical instability. Once the saturation of this instability is reached, we observe a turbulent state for which energy is supplied through the resonant inertial waves only. Depending on the amplitude of the rotation rate modulation, two different saturation states are observed. At large forcing amplitudes, the saturation flow main feature is a steady, geostrophic anticyclone. Its amplitude vanishes as the forcing amplitude is decreased while remaining above the threshold of the elliptical instability. Below this secondary transition, the saturation flow is a superposition of inertial waves which are in weakly non-linear resonant interaction, a state that could asymptotically lead to inertial wave turbulence. In addition to being a first experimental observation of a wave-dominated saturation in unstable rotating flows, the present study is also an experimental confirmation of the model of Le Reun et al, PRL 2017 who introduced the possibility of these two turbulent regimes. The transition between these two regimes and there relevance to geophysical applications are finally discussed.

Laurent Limat

## Circular hydraulic jump and inclined jump

Laurent Limat
laurent.limat@univ-paris-diderot.fr
Laboratoire Matière et Systèmes Complexes, Paris

We have investigated the flow and interface structures involved in a circular hydraulic jump formed by impacting a large horizontal disk with a jet of viscous liquid. Among other results, we found that the Froude number at the jump entry seems to be locked to a critical, constant value. This empirical condition, when combined with the large scale lubrication flow structure leads to a “à la Bohr” scaling, with Logarithmic corrections that can be explicitly calculated, in agreement with recent theoretical and numerical modeling. In a second step, we have investigated the jump structure formed when the jet and the impacted disk are inclined of the same amount, after varying the wetting conditions on the disk (hydrophilic, partial wetting and superhydrophobic). The results are very sensitive to the wetting properties as well as to the flow rate and plate inclination. We have tried to interpret the scaling laws observed with simple models generalizing Watson approach of the circular hydraulic jump.

Paul Milewski

## Statistics and models for Faraday pilot waves

Paul Milewski
p.a.milewski@bath.ac.uk
Department of Mathematical Sciences, University of Bath
Faraday pilot waves are a newly discovered hydrodynamic structure that consists a bouncing droplet which creates, and is propelled by, a Faraday wave. These pilot waves can behave in extremely complex ways exhibiting a classical form of wave-particle duality, and result in dynamics mimicking quantum mechanics, including multiple quantisation and probabilistic particle distributions reminiscent of QM. I will show a simple surface wave-droplet fluid model derived from the fluid equations, that captures many of the features observed, and focus on rationalising the emergence of the statistics of complex states and on building models describing particle statistics.

Nicolas Mordant

## Wave turbulence at the surface of water: the role of bound waves on intermittency

Nicolas Mordant
nicolas.mordant@univ.grenoble-alpes.fr
LEGI, Univesité Grenoble Alpes, CNRS, Grenoble-INP, Grenoble, France
By using a stereoscopic imaging technique, we could obtain a space-time resolved measurement of wave turbulence at the surface of water in a 13-m diameter tank. Wave are excited by meter-sized wedge wave makers that are close to omnidirectional. A frequency-wavenumber analysis shows that a turbulent regime develops that is made of a superposition of free waves and bound waves as expected for gravity surface waves. These bound waves result from triadic nonlinear interaction that provide energy to Fourier modes that are not lying on the linear dispersion relation (and thus non resonant). By performing a filtering in the Fourier space, we can remove the bound wave contribution to keep only the free wave one and we show, first, that the observed weak turbulence is indeed weakly nonlinear and, second, that the filtered field is much closer to Gaussian statistics. Furthermore the observed intermittency is strongly reduced so that the free-wave field is close to Gaussian at all scales.

Fernando Peruani

## Bacteria display optimal transport near surfaces

Fernando Peruani
peruani@unice.fr
Laboratoire J.A. Dieudonné, Nice
The near-surface swimming patterns of bacteria are determined by hydrodynamic interactions between the bacteria and the surface, which trap the bacteria in smooth circular trajectories that lead to inefficient surface exploration. Here, we combine experiments with a data-driven mathematical model to show that the surface exploration of a pathogenic strain of Escherichia coli results from a complex interplay between motility and transient surface adhesion events. These events allow the bacteria to break the smooth circular trajectories and regulate their transport properties by exploiting stop events that are facilitated by surface adhesion and lead to characteristic intermittent motion on surfaces. We find that the experimentally measured frequency of these stop-adhesion events coincides with the value that maximizes bacterial surface diffusivity according to our mathematical model. We discuss the applicability of our experimental and theoretical results to other bacterial strains on different surfaces. Our findings suggest that swimming bacteria use transient adhesion as a generic mechanism to regulate surface motion.

Yves Pomeau

## Intermittency and Leray singularities

Yves Pomeau
pomeau@lps.ens.fr
Real turbulent flows display intense and short lived velocity fluctuations. This is the phenomenon of intermittency. In 1934 Leray introduced the idea of finite time singularities of the incompressible fluid equations with smooth initial data. Leray singularities occur at given points of space and time. Their analysis give a precise relation between the large velocity and the large acceleration one should observe, a result opposite to the prediction of K41 scaling laws. Leray-like scaling are amazingly well verified in the velocity records of turbulence measured in Modane wind tunnel. I'll introduce Leray's idea and make the connection with the presentation by Christophe Josserand at this conference.

Giuseppe Pucci

## Spin lattices of walking droplets

Giuseppe Pucci
giuseppe.pucci@univ-rennes1.fr
Univ Rennes, CNRS, IPR (Institut de Physique de Rennes) ­ UMR 6251, F­35000 Rennes, France

A droplet bouncing on the surface of a vibrating liquid bath can self-propel across the surface through interaction with the wave field it generates by bouncing. These walking droplets or “walkers” comprise a droplet and its guiding wave, and have been shown to exhibit several behaviors analog to quantum systems. Most analogs consider a single walker interacting with boundaries or experiencing external forces. Controlling multiple walkers is challenging as their continuous wave-mediated interactions usually lead to pair bound states and droplet-droplet coalescence. Here I show that multiple walkers can be manipulated by designing the bottom topography of the vibrating bath as a lattice composed of deeper regions separated by shallow regions. Specifically, I show that circular wells at the bottom of the fluid bath encourage individual droplets to walk in clockwise or counter-clockwise direction along circular trajectories centered at the lattice sites. A thin fluid layer between the wells enables wave-mediated interactions between neighboring walkers resulting in ordered rotation dynamics across the lattice. When the pair coupling is sufficiently strong, interactions between neighboring droplets may induce local spin flips leading to ferromagnetic or anti-ferromagnetic order. In addition, an anti-ferromagnetic to ferromagnetic transition is obtained when the whole bath is rotating. Our experiments demonstrate the spontaneous emergence of collective behavior of walkers that mimic spin lattices.

This work has been done at Massachusetts Institute of Technology with Pedro J. Saenz, Sam E. Turton, Alexis Goujon, Rodolfo R. Rosales, Jörn Dunkel and John W. M. Bush.

Marc Rabaud

## When wind waves become Francis solitons

Marc Rabaud
rabaud@fast.u-psud.fr
Laboratoire FAST, Univ. Paris-Sud, Orsay.
When wind blows above a liquid surface, wind-waves form. We will show that if the liquid is viscous enough these waves becomes strongly non-linear solitary waves.

Christophe Raufaste

## Nonlinear waves in Plateau borders

Christophe Raufaste
christophe.raufaste@unice.fr
Institut de Physique de Nice, Nice

Plateau borders are the liquid microchannels found at the intersection between bubbles inside liquid foams. They concentrate most of the mass and their role is essential to account for the foams drainage and mechanical properties. During this presentation, experiments and results will be shown about the relaxation of a single Plateau border that is subject to external perturbations. We will see how a negative effective surface tension drives the dynamics, with a special emphasis on regimes dominated by inertial flows and nonlinear waves.

Christian Ruyer-Quil

## Sheared falling film flows: a numerical study

Christian Ruyer-Quil
christian.ruyer-quil@univ-smb.fr
Université de Savoie Mont-Blanc (Chambéry)
This work is devoted to the analysis of a counter-current gas-liquid film flow in an inclined channel. Our purpose is to consider how travelling waves generated at the free surface of the film by the classical Kapitza instability mechanism are affected by the gaseous turbulent flow. We aim at reproducing the experimental results obtained by Sophie mergui and Nicolas Kofman at FAST laboratory. Travelling wave solutions, i.e. waves which remain stationary in their moving frame, have been found numerically. The approach is one-sided as the interfacial stress is a function of the interface position only (we use Camassa's model to compute the shear exerted by the gas flow onto the liquid). A pseudo-spectral approach is followed where a projection of the unknowns onto Chebyshev polynomial functions is performed and the primitive equations are evaluated at the Gauss-Lobatto points, which results in the elimination of the normal coordinate. By invoking a penalization method to account for the continuity of the stresses at the free surface, an autonomous dynamical system of large dimension is obtained. Travelling wave solutions are then obtained as Hopf bifurcations of the Nusselt flat-film solution by means of a predictor-corrector continuation method (ATO07p software. A good agreement is found at a relatively low superficial gas speed. Though only qualitative, the proposed one-sided modelling is able to retrieve qualitatively the experimental observations: enhancement of wave amplitude, lowering of phase velocity and reduction of the number of capillary ripples.

Jesus Sanchez Rodriguez

## A simplified model of aquatic locomotion

Jesus Sanchez_Rodriguez
jesus.sanchez@inphyni.cnrs.fr
UCA, INPHYNI, CNRS, 1361 route des lucioles, 06560 Valbonne, France

We have developed a simple model of aquatic locomotion. Using the theory of complex variables, we have estimated the hydrodynamic forces acting on an infinite thin rigid plate of length L, following the seminal Work of Theordorsen [1].

By considering the different possible motions of the swimmer, we calculate the velocity potential to derive the pressure by means of the generalised Bernoulli relation. We show that the effect of flow unsteadiness is the principal mechanism for locomotion [2].

We impose a periodic rotation of the tail in order to approximate the undulatory motion of the swimmer. We show the linear dependence of longitudinal velocity on the angular frequency predicted by Gazzola et al [3] . We also predict that the transverse motion presents the same frequency as the forcing whereas the longitudinal motion is a linear function of time plus a periodic term with double frequency.

Finally, by taking the angle of the tail as a small parameter we perform a perturbative expansion to obtain an equation linking swimming velocity to the different parameters involved in swimming. The results arised from this perturbative method are in high accordance with the numerical results.

[1] Theodorsen, T., General theory of aerodynamic instability and the mechanism of flutter, NACA TR No. 496, 1934

[2] Garrick, I. E., Propulsion of a flapping and oscillating airfoil, NACA TR No. 567, 1936

[3] Gazzola, M., Argentina, M., & Mahadevan, L , Scaling macroscopic aquatic locomotion, Nature Physics 10 (10), 758-761, 2014

Benjamin Thiria

## Interference Model for an Array of Wave-Energy-Absorbing Flexible Structures

Benjamin Thiria
benjamin.thiria@espci.fr
PMMH-ESPCI, Paris, France
The present work focuses on the local effects of wave-structure interactions within an array of oscillating absorbers to optimize global effects, such as reflection, damping, and energy absorption. We use a model system of flexible blades, subjected to monochromatic waves, and develop a simplified one-dimensional model to predict optimal configurations, depending on various parameters, which include the number of blades, their spacing, and their flexibility. Optimal configurations are found to be close to regular patterns, and the impact of array configurations is shown to be limited regarding wave dissipation, mainly due to a competition between reflection and absorption.